Find positive numbers $a$ and $b$ so that the change of variables $s=ax, t=by$ transforms the integral $\int\int_R \,dx\,dy$ into $$\int\int_T \left\vert \frac{\partial(x,y)}{\partial(s,t)}\right\vert \,ds\,dt$$ for the region $R$, the rectangle $0\le x \le 20$, $0\le y \le 50$ and the region $T$, the square $0\le s, t \le 1$.

$a =$
$b =$

What is $\left\vert \frac{\partial(x,y)}{\partial(s,t)}\right\vert$ in this case?
$\left\vert \frac{\partial(x,y)}{\partial(s,t)}\right\vert =$